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Artificial intelligence combined with nonlinear optimization techniques and their application for yield curve optimization
1. | Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran |
2. | Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran |
This study makes use of the artificial intelligence approaches combined with some nonlinear optimization techniques for optimization of a well-known problem in financial engineering called yield curve. Yield curve estimation plays an important role on making strategic investment decisions. In this paper, we use two well-known parsimonious estimation models, Nelson-Siegel and Extended Nelson-Siegel, for the yield curve estimation. The proposed models of this paper are formulated as continuous nonlinear optimization problems. The resulted models are then solved using some nonlinear optimization and meta-heuristic approaches. The optimization techniques include hybrid GPSO parallel trust region-dog leg, Hybrid GPSO parallel trust region-nearly exact, Hybrid GPSO parallel Levenberg-Marquardt and Hybrid genetic electromagnetism like algorithm. The proposed models of this paper are examined using some real-world data from the bank of England and the results are analyzed.
References:
[1] |
I. Baki, Yield curve estimation by spline-based model, A thesis submitted to the Graduate school of Applied Mathematics of The Middle East Technical University, 2006. |
[2] |
M. S. Bazaraa, H. D. Sherali and C. M. Shetty,
Nonlinear Programming: Theory and Algorithms, 3$^{rd}$ edition, John Wiley and Sons, 2006.
doi: 10.1002/0471787779. |
[3] |
S. I. Birbil and S. C. Fang,
An electromagnetism-like mechanism for global optimization, Journal of Global Optimization, 25 (2003), 263-282.
doi: 10.1023/A:1022452626305. |
[4] |
R. R. Bliss, Testing term structure estimation methods, Working Paper WP 96-12a, Federal Reserve Bank of Atlanta, 1996. |
[5] |
D. J. Bolder and D. Streliski,
Yield Curve Modeling at the Bank of Canada, Technical Report TR 84, Bank of Canada, 1999. |
[6] |
M. Clerc and J. Kennedy,
The particle swarm explosion stability and convergence in a multi dimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2001), 58-73.
|
[7] |
A. Csajbok, Zero-coupon yield curve estimation from a central bank perspective, Working Paper WP 1998-2, National Bank of Hungary, 1998. |
[8] |
F. X. Diebold and C. Li,
Forecasting the term structure of government bond yields, Journal of Econometrics, 130 (2006), 337-364.
doi: 10.1016/j.jeconom.2005.03.005. |
[9] |
R. C. Eberhart and Y. Shi,
Comparing inertia weights and constriction factors in particle swarm optimization, Proceedings of the Congress on Evolutionary Computation, San Diego. CA, (2000), 84-88.
doi: 10.1109/CEC.2000.870279. |
[10] |
A. Hanjoori, M. Amiri and A. Alimi,
Forecasting stock price using grey-fuzzy technique and portfolio optimization by invasive weed optimization algorithm, Decision Science Letters, 2 (2013), 175-184.
|
[11] |
M. Ioannides,
A comparison of yield curve estimation techniques using UK data, Journal of Banking and Finance, 27 (2003), 1-26.
doi: 10.1016/S0378-4266(01)00217-5. |
[12] |
J. Kennedy and R. C. Eberhart,
Particle swarm optimization, Proceeding of IEEE International Conference on Neural Networks, (1995), 1942-1948.
doi: 10.1109/ICNN.1995.488968. |
[13] |
P. Manousopoulos and M. Michalopoulos,
A comparison of yield curve estimation methods: The Greek case, Journal of Financial Decision Making, 1 (2005), 33-46.
|
[14] |
P. Manousopoulos and M. Michalopoulos, Yield curve construction as a non-linear optimization problem, Proceedings of the 18th Hellenic Conference on Operations Research, 2006. |
[15] |
P. Manousopoulos and M. Michalopoulos,
Comparison of non-linear optimization algorithms for yield curve estimation, European Journal of Operational Research, 192 (2009), 594-602.
doi: 10.1016/j.ejor.2007.09.017. |
[16] |
J. H. McCulloch,
Measuring the term structure of interest rates, Journal of Business, 44 (1971), 19-31.
doi: 10.1086/295329. |
[17] |
J. H. McCulloch,
The tax-adjusted yield curve, The Journal of Finance, 30 (1975), 811-830.
doi: 10.1111/j.1540-6261.1975.tb01852.x. |
[18] |
C. R. Nelson and A. F. Siegel,
Parsimonious modeling of yield curves, The Journal of Business, 60 (1987), 473-489.
doi: 10.1086/296409. |
[19] |
J. Nocedal and S. J. Write,
Numerical Optimization, Springer Science and Business Media, 2006.
doi: MR2244940. |
[20] |
M. Orouji,
Oil price shocks and stock market returns, Accounting, 2 (2016), 103-108.
doi: 10.5267/j.ac.2016.2.005. |
[21] |
M. Pooter,
Examining the Nelson-Siegel Class of Term Structure Models, Tinbergen Institute, Discussion paper, 2007. |
[22] |
P. Soderlind and L. E. O. Svensson,
New Techniques to Extract Market Expectations from Financial Instruments, Working Paper WP 5877, National Bureau of Economic Research, 1996.
doi: 10.3386/w5877. |
[23] |
L. E. O. Svensson,
Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, Working Paper WP 4871, National Bureau of EconomicResearch, (1994).
doi: 10.3386/w4871. |
[24] |
L. E. O. Svensson,
Estimating forward interest rates with the Extended Nelson-Siegel method, Sveriges Riksbank Quarterly Review, 3 (1995), 13-26.
|
show all references
References:
[1] |
I. Baki, Yield curve estimation by spline-based model, A thesis submitted to the Graduate school of Applied Mathematics of The Middle East Technical University, 2006. |
[2] |
M. S. Bazaraa, H. D. Sherali and C. M. Shetty,
Nonlinear Programming: Theory and Algorithms, 3$^{rd}$ edition, John Wiley and Sons, 2006.
doi: 10.1002/0471787779. |
[3] |
S. I. Birbil and S. C. Fang,
An electromagnetism-like mechanism for global optimization, Journal of Global Optimization, 25 (2003), 263-282.
doi: 10.1023/A:1022452626305. |
[4] |
R. R. Bliss, Testing term structure estimation methods, Working Paper WP 96-12a, Federal Reserve Bank of Atlanta, 1996. |
[5] |
D. J. Bolder and D. Streliski,
Yield Curve Modeling at the Bank of Canada, Technical Report TR 84, Bank of Canada, 1999. |
[6] |
M. Clerc and J. Kennedy,
The particle swarm explosion stability and convergence in a multi dimensional complex space, IEEE Transactions on Evolutionary Computation, 6 (2001), 58-73.
|
[7] |
A. Csajbok, Zero-coupon yield curve estimation from a central bank perspective, Working Paper WP 1998-2, National Bank of Hungary, 1998. |
[8] |
F. X. Diebold and C. Li,
Forecasting the term structure of government bond yields, Journal of Econometrics, 130 (2006), 337-364.
doi: 10.1016/j.jeconom.2005.03.005. |
[9] |
R. C. Eberhart and Y. Shi,
Comparing inertia weights and constriction factors in particle swarm optimization, Proceedings of the Congress on Evolutionary Computation, San Diego. CA, (2000), 84-88.
doi: 10.1109/CEC.2000.870279. |
[10] |
A. Hanjoori, M. Amiri and A. Alimi,
Forecasting stock price using grey-fuzzy technique and portfolio optimization by invasive weed optimization algorithm, Decision Science Letters, 2 (2013), 175-184.
|
[11] |
M. Ioannides,
A comparison of yield curve estimation techniques using UK data, Journal of Banking and Finance, 27 (2003), 1-26.
doi: 10.1016/S0378-4266(01)00217-5. |
[12] |
J. Kennedy and R. C. Eberhart,
Particle swarm optimization, Proceeding of IEEE International Conference on Neural Networks, (1995), 1942-1948.
doi: 10.1109/ICNN.1995.488968. |
[13] |
P. Manousopoulos and M. Michalopoulos,
A comparison of yield curve estimation methods: The Greek case, Journal of Financial Decision Making, 1 (2005), 33-46.
|
[14] |
P. Manousopoulos and M. Michalopoulos, Yield curve construction as a non-linear optimization problem, Proceedings of the 18th Hellenic Conference on Operations Research, 2006. |
[15] |
P. Manousopoulos and M. Michalopoulos,
Comparison of non-linear optimization algorithms for yield curve estimation, European Journal of Operational Research, 192 (2009), 594-602.
doi: 10.1016/j.ejor.2007.09.017. |
[16] |
J. H. McCulloch,
Measuring the term structure of interest rates, Journal of Business, 44 (1971), 19-31.
doi: 10.1086/295329. |
[17] |
J. H. McCulloch,
The tax-adjusted yield curve, The Journal of Finance, 30 (1975), 811-830.
doi: 10.1111/j.1540-6261.1975.tb01852.x. |
[18] |
C. R. Nelson and A. F. Siegel,
Parsimonious modeling of yield curves, The Journal of Business, 60 (1987), 473-489.
doi: 10.1086/296409. |
[19] |
J. Nocedal and S. J. Write,
Numerical Optimization, Springer Science and Business Media, 2006.
doi: MR2244940. |
[20] |
M. Orouji,
Oil price shocks and stock market returns, Accounting, 2 (2016), 103-108.
doi: 10.5267/j.ac.2016.2.005. |
[21] |
M. Pooter,
Examining the Nelson-Siegel Class of Term Structure Models, Tinbergen Institute, Discussion paper, 2007. |
[22] |
P. Soderlind and L. E. O. Svensson,
New Techniques to Extract Market Expectations from Financial Instruments, Working Paper WP 5877, National Bureau of Economic Research, 1996.
doi: 10.3386/w5877. |
[23] |
L. E. O. Svensson,
Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994, Working Paper WP 4871, National Bureau of EconomicResearch, (1994).
doi: 10.3386/w4871. |
[24] |
L. E. O. Svensson,
Estimating forward interest rates with the Extended Nelson-Siegel method, Sveriges Riksbank Quarterly Review, 3 (1995), 13-26.
|










Methods | ||||||
Models | Nelson-Siegel | Extended Nelson-Siegel | ||||
Average | Standard deviation | Average Time (in minutes) | Average deviation | Standard | Average Time (in minutes) | |
HGEM | 0.103029 | 0.020578 | 1.607608 | 0.007307 | 0.002806582 | 2.761143 |
HGPSO | 0.103976 | 0.01973 | 1.05098 | 0.012025 | 0.006613485 | 1.89916 |
HGPSOPTR_NE | 0.102208 | 0.023577 | 29.1853 | 0.004866667 | 0.002573407 | 36.364 |
HGPSOPTR_DL | 0.102167 | 0.023679 | 31.3681 | 0.005108333 | 0.002877644 | 37.85 |
HGPSOPLM | 0.102533 | 0.006366 | 27.4275 | 0.004783333 | 0.002648785 | 35.74 |
Methods | ||||||
Models | Nelson-Siegel | Extended Nelson-Siegel | ||||
Average | Standard deviation | Average Time (in minutes) | Average deviation | Standard | Average Time (in minutes) | |
HGEM | 0.103029 | 0.020578 | 1.607608 | 0.007307 | 0.002806582 | 2.761143 |
HGPSO | 0.103976 | 0.01973 | 1.05098 | 0.012025 | 0.006613485 | 1.89916 |
HGPSOPTR_NE | 0.102208 | 0.023577 | 29.1853 | 0.004866667 | 0.002573407 | 36.364 |
HGPSOPTR_DL | 0.102167 | 0.023679 | 31.3681 | 0.005108333 | 0.002877644 | 37.85 |
HGPSOPLM | 0.102533 | 0.006366 | 27.4275 | 0.004783333 | 0.002648785 | 35.74 |
Variables | ||||||||||||
Methods | | | | | | | ||||||
Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | |
HGEM | 9.496 | 0.384 | -8.982 | 0.378 | -6.656 | 0.142 | -12.387 | 1.031 | 2.482 | 0.061 | 23.948 | 0.263 |
HGPSO | 9.470 | 0.350 | -8.978 | 0.381 | -6.585 | 0.035 | -11.786 | 0.471 | 2.480 | 0.078 | 24.239 | 0.144 |
HGPSOPTR-NE | 9.796 | 1.007 | -9.324 | 1.035 | -6.688 | 0.549 | -13.333 | 2.883 | 2.571 | 0.162 | 24.619 | 1.110 |
HGPSOPTR-DL | 9.917 | 0.849 | -9.448 | 0.866 | -6.738 | 0.531 | -13.666 | 2.442 | 2.586 | 0.117 | 24.462 | 1.286 |
HGPSOPLM | 9.957 | 0.850 | -9.491 | 0.873 | -6.725 | 0.574 | -13.796 | 2.469 | 2.604 | 0.111 | 24.610 | 1.119 |
Variables | ||||||||||||
Methods | | | | | | | ||||||
Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | |
HGEM | 9.496 | 0.384 | -8.982 | 0.378 | -6.656 | 0.142 | -12.387 | 1.031 | 2.482 | 0.061 | 23.948 | 0.263 |
HGPSO | 9.470 | 0.350 | -8.978 | 0.381 | -6.585 | 0.035 | -11.786 | 0.471 | 2.480 | 0.078 | 24.239 | 0.144 |
HGPSOPTR-NE | 9.796 | 1.007 | -9.324 | 1.035 | -6.688 | 0.549 | -13.333 | 2.883 | 2.571 | 0.162 | 24.619 | 1.110 |
HGPSOPTR-DL | 9.917 | 0.849 | -9.448 | 0.866 | -6.738 | 0.531 | -13.666 | 2.442 | 2.586 | 0.117 | 24.462 | 1.286 |
HGPSOPLM | 9.957 | 0.850 | -9.491 | 0.873 | -6.725 | 0.574 | -13.796 | 2.469 | 2.604 | 0.111 | 24.610 | 1.119 |
Variables | ||||||||
Methods | | | | | ||||
Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | |
HGEM | 5.2881 | 0.0004 | -3.8043 | 0.0053 | -5.4705 | 0.0093 | 1.5192 | 0.0013 |
HGPSO | 5.2467 | 0.0919 | -4.5282 | 0.0732 | -5.2269 | 0.5462 | 1.6515 | 0.2967 |
HGPSOPTR-NE | 5.0321 | 0.0581 | -4.3450 | 0.0458 | -4.1469 | 0.0422 | 2.2402 | 0.0073 |
HGPSOPTR-DL | 5.0028 | 0.9806 | -4.3148 | 0.8265 | -4.0892 | 4.9073 | 2.273 | 2.6319 |
HGPSOPLM | 5.0832 | 0.0581 | -4.3976 | 0.0458 | -4.2517 | 0.0422 | 2.1824 | 0.0072 |
Variables | ||||||||
Methods | | | | | ||||
Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | Average | Standard deviation | |
HGEM | 5.2881 | 0.0004 | -3.8043 | 0.0053 | -5.4705 | 0.0093 | 1.5192 | 0.0013 |
HGPSO | 5.2467 | 0.0919 | -4.5282 | 0.0732 | -5.2269 | 0.5462 | 1.6515 | 0.2967 |
HGPSOPTR-NE | 5.0321 | 0.0581 | -4.3450 | 0.0458 | -4.1469 | 0.0422 | 2.2402 | 0.0073 |
HGPSOPTR-DL | 5.0028 | 0.9806 | -4.3148 | 0.8265 | -4.0892 | 4.9073 | 2.273 | 2.6319 |
HGPSOPLM | 5.0832 | 0.0581 | -4.3976 | 0.0458 | -4.2517 | 0.0422 | 2.1824 | 0.0072 |
Method | Data | Eigenvalues | Gnorm | |||||
Set1 | 219.717 | 7.245 | 2.530 | 0.041 | 0 | 0.0009 | 0.0004 | |
Set2 | 220.752 | 7.111 | 2.748 | 0.042 | 0.0001 | 0.001 | 0.011 | |
Set3 | 214.848 | 6.942 | 2.733 | 0.041 | 0 | 0.0009 | 0.006 | |
Set4 | 188.925 | 5.999 | 3.557 | 0.046 | 0.001 | 0.0006 | 0.026 | |
Set5 | 198.835 | 6.254 | 3.134 | 0.072 | -0.005 | 0.003 | 0.861 | |
Set6 | 184.056 | 5.863 | 2.929 | 0.039 | 0.0002 | 0.0006 | 0.003 | |
HGPSOPTR-NE | Set7 | 220.411 | 6.617 | 3.233 | 0.043 | 0 | 0.002 | 0.041 |
Set8 | 201.848 | 6.490 | 2.754 | 0.039 | 0 | 0.0007 | 0.002 | |
Set9 | 201.782 | 6.450 | 2.712 | 0.047 | -0.002 | 0.002 | 0.299 | |
Set10 | 192.970 | 6.112 | 2.971 | 0.040 | 0.0007 | 0.0001 | 0.002 | |
Set11 | 192.574 | 6.087 | 3.086 | 0.043 | -0.001 | 0.001 | 0.374 | |
Set12 | 206.778 | 6.637 | 2.896 | 0.042 | 0.0001 | 0.0009 | 0.004 | |
Average | 203.624 | 6.484 | 2.940 | 0.045 | -0.0005 | 0.0012 | 0.136 | |
Set1 | 215.781 | 7.093 | 2.624 | 0.042 | 0 | 0.0009 | 0.009 | |
Set2 | 217.899 | 7.074 | 2.697 | 0.042 | 0 | 0.0009 | 0.0008 | |
Set3 | 210.860 | 6.806 | 2.826 | 0.042 | 0 | 0.0009 | 0.0137 | |
Set4 | 208.092 | 6.804 | 2.728 | 0.041 | 0 | 0.0008 | 0.009 | |
Set5 | 198.835 | 6.254 | 3.134 | 0.072 | -0.005 | 0.003 | 0.861 | |
Set6 | 180.292 | 5.633 | 3.236 | 0.042 | 0.0004 | 0.0007 | 0.009 | |
HGPSOPTR-DL | Set7 | 220.411 | 6.617 | 3.233 | 0.043 | 0 | 0.002 | 0.041 |
Set8 | 216.129 | 6.706 | 3.019 | 0.041 | 0.0001 | 0.002 | 0.014 | |
Set9 | 194.599 | 6.178 | 2.895 | 0.039 | 0.0001 | 0.0007 | 0.007 | |
Set10 | 195.845 | 6.264 | 2.811 | 0.039 | 0 | 0.0007 | 0.0121 | |
Set11 | 195.095 | 6.153 | 2.995 | 0.040 | 0 | 0.0007 | 0.0002 | |
Set12 | 207.949 | 6.682 | 2.881 | 0.042 | 0 | 0.0009 | 0.002 | |
Average | 205.149 | 6.522 | 2.923 | 0.044 | -0.0004 | 0.001 | 0.0815 | |
Set1 | 225.272 | 7.332 | 2.588 | 0.042 | 0 | 0.001 | 0.003 | |
Set2 | 217.686 | 7.065 | 2.709 | 0.042 | 0 | 0.0009 | 0.005 | |
Set3 | 212.355 | 6.863 | 2.781 | 0.042 | 0 | 0.0009 | 0.001 | |
Set4 | 210.691 | 6.901 | 2.684 | 0.042 | 0 | 0.0008 | 0.004 | |
Set5 | 198.835 | 6.254 | 3.134 | 0.072 | -0.005 | 0.003 | 0.861 | |
Set6 | 184.161 | 5.865 | 2.933 | 0.039 | 0.0002 | 0.0006 | 0.008 | |
HGPSOPLM | Set7 | 220.411 | 6.617 | 3.233 | 0.043 | 0 | 0.002 | 0.041 |
Set8 | 201.916 | 6.494 | 2.751 | 0.039 | 0 | 0.0007 | 0.007 | |
Set9 | 195.245 | 6.211 | 2.868 | 0.039 | 0 | 0.0007 | 0.003 | |
Set10 | 192.97 | 6.112 | 2.971 | 0.04 | 0.0001 | 0.0007 | 0.002 | |
Set11 | 192.575 | 6.088 | 3.086 | 0.043 | -0.001 | 0.001 | 0.374 | |
Set12 | 206.734 | 6.636 | 2.895 | 0.042 | 0 | 0.0009 | 0.002 | |
Average | 204.904 | 6.536 | 2.886 | 0.044 | -0.0005 | 0.001 | 0.109 |
Method | Data | Eigenvalues | Gnorm | |||||
Set1 | 219.717 | 7.245 | 2.530 | 0.041 | 0 | 0.0009 | 0.0004 | |
Set2 | 220.752 | 7.111 | 2.748 | 0.042 | 0.0001 | 0.001 | 0.011 | |
Set3 | 214.848 | 6.942 | 2.733 | 0.041 | 0 | 0.0009 | 0.006 | |
Set4 | 188.925 | 5.999 | 3.557 | 0.046 | 0.001 | 0.0006 | 0.026 | |
Set5 | 198.835 | 6.254 | 3.134 | 0.072 | -0.005 | 0.003 | 0.861 | |
Set6 | 184.056 | 5.863 | 2.929 | 0.039 | 0.0002 | 0.0006 | 0.003 | |
HGPSOPTR-NE | Set7 | 220.411 | 6.617 | 3.233 | 0.043 | 0 | 0.002 | 0.041 |
Set8 | 201.848 | 6.490 | 2.754 | 0.039 | 0 | 0.0007 | 0.002 | |
Set9 | 201.782 | 6.450 | 2.712 | 0.047 | -0.002 | 0.002 | 0.299 | |
Set10 | 192.970 | 6.112 | 2.971 | 0.040 | 0.0007 | 0.0001 | 0.002 | |
Set11 | 192.574 | 6.087 | 3.086 | 0.043 | -0.001 | 0.001 | 0.374 | |
Set12 | 206.778 | 6.637 | 2.896 | 0.042 | 0.0001 | 0.0009 | 0.004 | |
Average | 203.624 | 6.484 | 2.940 | 0.045 | -0.0005 | 0.0012 | 0.136 | |
Set1 | 215.781 | 7.093 | 2.624 | 0.042 | 0 | 0.0009 | 0.009 | |
Set2 | 217.899 | 7.074 | 2.697 | 0.042 | 0 | 0.0009 | 0.0008 | |
Set3 | 210.860 | 6.806 | 2.826 | 0.042 | 0 | 0.0009 | 0.0137 | |
Set4 | 208.092 | 6.804 | 2.728 | 0.041 | 0 | 0.0008 | 0.009 | |
Set5 | 198.835 | 6.254 | 3.134 | 0.072 | -0.005 | 0.003 | 0.861 | |
Set6 | 180.292 | 5.633 | 3.236 | 0.042 | 0.0004 | 0.0007 | 0.009 | |
HGPSOPTR-DL | Set7 | 220.411 | 6.617 | 3.233 | 0.043 | 0 | 0.002 | 0.041 |
Set8 | 216.129 | 6.706 | 3.019 | 0.041 | 0.0001 | 0.002 | 0.014 | |
Set9 | 194.599 | 6.178 | 2.895 | 0.039 | 0.0001 | 0.0007 | 0.007 | |
Set10 | 195.845 | 6.264 | 2.811 | 0.039 | 0 | 0.0007 | 0.0121 | |
Set11 | 195.095 | 6.153 | 2.995 | 0.040 | 0 | 0.0007 | 0.0002 | |
Set12 | 207.949 | 6.682 | 2.881 | 0.042 | 0 | 0.0009 | 0.002 | |
Average | 205.149 | 6.522 | 2.923 | 0.044 | -0.0004 | 0.001 | 0.0815 | |
Set1 | 225.272 | 7.332 | 2.588 | 0.042 | 0 | 0.001 | 0.003 | |
Set2 | 217.686 | 7.065 | 2.709 | 0.042 | 0 | 0.0009 | 0.005 | |
Set3 | 212.355 | 6.863 | 2.781 | 0.042 | 0 | 0.0009 | 0.001 | |
Set4 | 210.691 | 6.901 | 2.684 | 0.042 | 0 | 0.0008 | 0.004 | |
Set5 | 198.835 | 6.254 | 3.134 | 0.072 | -0.005 | 0.003 | 0.861 | |
Set6 | 184.161 | 5.865 | 2.933 | 0.039 | 0.0002 | 0.0006 | 0.008 | |
HGPSOPLM | Set7 | 220.411 | 6.617 | 3.233 | 0.043 | 0 | 0.002 | 0.041 |
Set8 | 201.916 | 6.494 | 2.751 | 0.039 | 0 | 0.0007 | 0.007 | |
Set9 | 195.245 | 6.211 | 2.868 | 0.039 | 0 | 0.0007 | 0.003 | |
Set10 | 192.97 | 6.112 | 2.971 | 0.04 | 0.0001 | 0.0007 | 0.002 | |
Set11 | 192.575 | 6.088 | 3.086 | 0.043 | -0.001 | 0.001 | 0.374 | |
Set12 | 206.734 | 6.636 | 2.895 | 0.042 | 0 | 0.0009 | 0.002 | |
Average | 204.904 | 6.536 | 2.886 | 0.044 | -0.0005 | 0.001 | 0.109 |
Method | Data | Eigenvalues | Gnorm | |||
Set 1 | 166.4771 | 6.6645 | 3.359 | 0.0499 | 0.0032 | |
Set 2 | 168.9473 | 6.8236 | 3.3593 | 0.0498 | 0.0069 | |
Set 3 | 167.3736 | 6.8091 | 3.3541 | 0.0489 | 0.0025 | |
Set 4 | 166.0604 | 6.5357 | 3.3699 | 0.0494 | 0.0033 | |
Set 5 | 161.8373 | 5.9944 | 3.4018 | 0.0482 | 0.0056 | |
Set 6 | 144.2963 | 3.0416 | 0.3247 | 0.0013 | 0.005 | |
HGPSOPTR-NE | Set 7 | 162.1993 | 5.941 | 3.4066 | 0.0456 | 0.0081 |
Set 8 | 164.1612 | 6.2443 | 3.3904 | 0.0477 | 0.0105 | |
Set 9 | 162.8706 | 6.1114 | 3.3942 | 0.0468 | 0.0054 | |
Set 10 | 162.2295 | 5.9982 | 3.4065 | 0.0473 | 0.0022 | |
Set 11 | 163.7136 | 6.2145 | 3.3885 | 0.0472 | 0.0009 | |
Set 12 | 167.1193 | 6.6272 | 3.3702 | 0.0495 | 0.0005 | |
Average | 163.1071 | 6.0838 | 3.1271 | 0.0443 | 0.0045 | |
Set1 | 166.4771 | 6.6645 | 3.359 | 0.0499 | 0.0032 | |
Set2 | 168.9473 | 6.8236 | 3.3593 | 0.0498 | 0.0069 | |
Set3 | 167.3736 | 6.8091 | 3.3541 | 0.0489 | 0.0025 | |
Set4 | 166.0604 | 6.5357 | 3.3699 | 0.0494 | 0.0033 | |
Set5 | 161.8373 | 5.9944 | 3.4018 | 0.0482 | 0.0056 | |
Set6 | 145.1737 | 2.9842 | 0.0011 | 0.3113 | 0.0065 | |
HGPSOPTR-DL | Set7 | 162.1993 | 5.941 | 3.4066 | 0.0456 | 0.0081 |
Set8 | 164.1612 | 6.2443 | 3.3904 | 0.0477 | 0.0105 | |
Set9 | 162.8699 | 6.1117 | 3.3942 | 0.0468 | 0.0035 | |
Set10 | 162.2295 | 5.9982 | 3.4065 | 0.0473 | 0.0022 | |
Set11 | 162.7722 | 5.7516 | 3.4069 | 0.02 | 1.969 | |
Set12 | 167.1193 | 6.6272 | 3.3702 | 0.0495 | 0.0005 | |
Average | 163.1017 | 6.0405 | 3.1017 | 0.0679 | 0.1685 | |
Set1 | 166.4771 | 6.6645 | 3.359 | 0.0499 | 0.0032 | |
Set2 | 168.9473 | 6.8236 | 3.3593 | 0.0498 | 0.0069 | |
Set3 | 168.368 | 6.8067 | 3.3543 | 0.0488 | 0.0066 | |
Set4 | 166.0697 | 6.5361 | 3.3699 | 0.0493 | 0.0069 | |
Set5 | 161.8373 | 5.9944 | 3.4018 | 0.0482 | 0.0056 | |
Set6 | 142.6002 | 3.1483 | 0.002 | 0.3515 | 0.0065 | |
HGPSOPLM | Set7 | 162.1993 | 5.941 | 3.4066 | 0.0456 | 0.0081 |
Set8 | 164.1612 | 6.2443 | 3.3904 | 0.0477 | 0.0105 | |
Set9 | 162.8699 | 6.1117 | 3.3942 | 0.0468 | 0.0035 | |
Set10 | 162.2295 | 5.9982 | 3.4065 | 0.0473 | 0.0022 | |
Set11 | 163.7136 | 6.2145 | 3.3885 | 0.0472 | 0.0009 | |
Set12 | 167.1193 | 6.6272 | 3.3702 | 0.0495 | 0.0005 | |
Average | 163.0494 | 6.0925 | 3.1002 | 0.0735 | 0.0051 |
Method | Data | Eigenvalues | Gnorm | |||
Set 1 | 166.4771 | 6.6645 | 3.359 | 0.0499 | 0.0032 | |
Set 2 | 168.9473 | 6.8236 | 3.3593 | 0.0498 | 0.0069 | |
Set 3 | 167.3736 | 6.8091 | 3.3541 | 0.0489 | 0.0025 | |
Set 4 | 166.0604 | 6.5357 | 3.3699 | 0.0494 | 0.0033 | |
Set 5 | 161.8373 | 5.9944 | 3.4018 | 0.0482 | 0.0056 | |
Set 6 | 144.2963 | 3.0416 | 0.3247 | 0.0013 | 0.005 | |
HGPSOPTR-NE | Set 7 | 162.1993 | 5.941 | 3.4066 | 0.0456 | 0.0081 |
Set 8 | 164.1612 | 6.2443 | 3.3904 | 0.0477 | 0.0105 | |
Set 9 | 162.8706 | 6.1114 | 3.3942 | 0.0468 | 0.0054 | |
Set 10 | 162.2295 | 5.9982 | 3.4065 | 0.0473 | 0.0022 | |
Set 11 | 163.7136 | 6.2145 | 3.3885 | 0.0472 | 0.0009 | |
Set 12 | 167.1193 | 6.6272 | 3.3702 | 0.0495 | 0.0005 | |
Average | 163.1071 | 6.0838 | 3.1271 | 0.0443 | 0.0045 | |
Set1 | 166.4771 | 6.6645 | 3.359 | 0.0499 | 0.0032 | |
Set2 | 168.9473 | 6.8236 | 3.3593 | 0.0498 | 0.0069 | |
Set3 | 167.3736 | 6.8091 | 3.3541 | 0.0489 | 0.0025 | |
Set4 | 166.0604 | 6.5357 | 3.3699 | 0.0494 | 0.0033 | |
Set5 | 161.8373 | 5.9944 | 3.4018 | 0.0482 | 0.0056 | |
Set6 | 145.1737 | 2.9842 | 0.0011 | 0.3113 | 0.0065 | |
HGPSOPTR-DL | Set7 | 162.1993 | 5.941 | 3.4066 | 0.0456 | 0.0081 |
Set8 | 164.1612 | 6.2443 | 3.3904 | 0.0477 | 0.0105 | |
Set9 | 162.8699 | 6.1117 | 3.3942 | 0.0468 | 0.0035 | |
Set10 | 162.2295 | 5.9982 | 3.4065 | 0.0473 | 0.0022 | |
Set11 | 162.7722 | 5.7516 | 3.4069 | 0.02 | 1.969 | |
Set12 | 167.1193 | 6.6272 | 3.3702 | 0.0495 | 0.0005 | |
Average | 163.1017 | 6.0405 | 3.1017 | 0.0679 | 0.1685 | |
Set1 | 166.4771 | 6.6645 | 3.359 | 0.0499 | 0.0032 | |
Set2 | 168.9473 | 6.8236 | 3.3593 | 0.0498 | 0.0069 | |
Set3 | 168.368 | 6.8067 | 3.3543 | 0.0488 | 0.0066 | |
Set4 | 166.0697 | 6.5361 | 3.3699 | 0.0493 | 0.0069 | |
Set5 | 161.8373 | 5.9944 | 3.4018 | 0.0482 | 0.0056 | |
Set6 | 142.6002 | 3.1483 | 0.002 | 0.3515 | 0.0065 | |
HGPSOPLM | Set7 | 162.1993 | 5.941 | 3.4066 | 0.0456 | 0.0081 |
Set8 | 164.1612 | 6.2443 | 3.3904 | 0.0477 | 0.0105 | |
Set9 | 162.8699 | 6.1117 | 3.3942 | 0.0468 | 0.0035 | |
Set10 | 162.2295 | 5.9982 | 3.4065 | 0.0473 | 0.0022 | |
Set11 | 163.7136 | 6.2145 | 3.3885 | 0.0472 | 0.0009 | |
Set12 | 167.1193 | 6.6272 | 3.3702 | 0.0495 | 0.0005 | |
Average | 163.0494 | 6.0925 | 3.1002 | 0.0735 | 0.0051 |
Average | Standard deviation | Average Time (in minutes) | |
HGEM | 0.024759 | 0.009674 | 3.0325 |
HGPSO | 0.039497 | 0.007324 | 1.1797 |
HGPSOPTR-NE | 0.016541 | 0.015271 | 35.3814 |
HGPSOPTR-DL | 0.019875 | 0.01700 | 39.1531 |
HGPSOPLM | 0.015778 | 0.014238 | 36.332 |
Average | Standard deviation | Average Time (in minutes) | |
HGEM | 0.024759 | 0.009674 | 3.0325 |
HGPSO | 0.039497 | 0.007324 | 1.1797 |
HGPSOPTR-NE | 0.016541 | 0.015271 | 35.3814 |
HGPSOPTR-DL | 0.019875 | 0.01700 | 39.1531 |
HGPSOPLM | 0.015778 | 0.014238 | 36.332 |
Random parameters as Initial condition | Yesterday parameters estimation as Initial condition | PSO results as Initial condition | |
Date | Final best value | Final best value | Final best value |
01/05/2008 | 0.5486 | - | 0.0368 |
02/05/2008 | 0.0540 | 0.4362 | 0.0081 |
06/05/2008 | 0.7640 | 0.5221 | 0.0489 |
07/05/2008 | 0.0534 | 0.4598 | 0.0084 |
08/05/2008 | 0.0536 | 0.6287 | 0.0144 |
09/05/2008 | 0.0550 | 0.6578 | 0.0166 |
12/05/2008 | 0.0126 | 0.7254 | 0.0672 |
13/05/2008 | 0.0705 | 0.6465 | 0.0064 |
14/05/2008 | 0.0825 | 0.4961 | 0.0037 |
15/05/2008 | 0.1040 | 0.4468 | 0.0346 |
16/05/2008 | 0.2365 | 0.7188 | 0.1200 |
19/05/2008 | 0.3837 | 0.4385 | 0.0742 |
20/05/2008 | 0.0914 | 0.4956 | 0.0023 |
21/05/2008 | 0.2751 | 0.3631 | 0.1220 |
22/05/2008 | 0.2099 | 0.2960 | 0.0838 |
23/05/2008 | 0.2504 | 0.4476 | 0.1214 |
27/05/2008 | 0.1863 | 0.3923 | 0.0768 |
28/05/2008 | 0.2189 | 0.3314 | 0.0869 |
29/05/2008 | 0.1662 | 0.1650 | 0.0880 |
30/05/2008 | 0.1637 | 0.1691 | 0.0846 |
Random parameters as Initial condition | Yesterday parameters estimation as Initial condition | PSO results as Initial condition | |
Date | Final best value | Final best value | Final best value |
01/05/2008 | 0.5486 | - | 0.0368 |
02/05/2008 | 0.0540 | 0.4362 | 0.0081 |
06/05/2008 | 0.7640 | 0.5221 | 0.0489 |
07/05/2008 | 0.0534 | 0.4598 | 0.0084 |
08/05/2008 | 0.0536 | 0.6287 | 0.0144 |
09/05/2008 | 0.0550 | 0.6578 | 0.0166 |
12/05/2008 | 0.0126 | 0.7254 | 0.0672 |
13/05/2008 | 0.0705 | 0.6465 | 0.0064 |
14/05/2008 | 0.0825 | 0.4961 | 0.0037 |
15/05/2008 | 0.1040 | 0.4468 | 0.0346 |
16/05/2008 | 0.2365 | 0.7188 | 0.1200 |
19/05/2008 | 0.3837 | 0.4385 | 0.0742 |
20/05/2008 | 0.0914 | 0.4956 | 0.0023 |
21/05/2008 | 0.2751 | 0.3631 | 0.1220 |
22/05/2008 | 0.2099 | 0.2960 | 0.0838 |
23/05/2008 | 0.2504 | 0.4476 | 0.1214 |
27/05/2008 | 0.1863 | 0.3923 | 0.0768 |
28/05/2008 | 0.2189 | 0.3314 | 0.0869 |
29/05/2008 | 0.1662 | 0.1650 | 0.0880 |
30/05/2008 | 0.1637 | 0.1691 | 0.0846 |
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